He'll be standing in the middle of an area scattered with corpses, and will go with the Chosen One over the Party Limit if failed a Luck check. His joining gives the Jinxed trait and drops Luck to 1 (or 2 for the gifted).

In battle, the pariah dog does nothing helpful and usually just runs away and over the edge of the screen.

The only way to remove the dog from the party and stop it from taking up one of the NPC slots is to kill it. When the pariah dog is removed from the party, the Jinxed trait vanishes and Luck returns to normal.

Zero damage critical hits that cause the death of any NPC with scripts that would normally activate upon death, will not activate those scripts (this bug is not limited to the pariah dog). Therefore, it is possible for the pariah dog to die, and all the ill effects of its presence in the party, including occupying a party slot, to remain. The perk Living Anatomy makes zero damage impossible; Bonus Ranged Damage and Bonus HtH Damage probably work the same way.